Question

3. simple random sample of 80 customers is taken from a customer information file and the average age is 34. The population standard deviation σ is unknown. Instead the sample standard deviation s is also calculated from the sample and is found to be 5.5. (6 points)
   
   1. Test the hypothesis that the population mean age is greater than 30 using the critical value approach and a 0.05 level of significance.
   2. Test the hypothesis that the population mean age is less than 37 using the p-value approach and a 0.05 level of significance.
   3. Test the hypothesis that the population mean age is different from 31 using the p-value approach and a 0.05 level of significance.

Answer 1

a)
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 30
Alternative Hypothesis, Ha: μ > 30

Rejection Region
This is right tailed test, for α = 0.05 and df = 79
Critical value of t is 1.664.
Hence reject H0 if t > 1.664

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (34 - 30)/(5.5/sqrt(80))
t = 6.505

reject the null hypothesis

b)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 37
Alternative Hypothesis, Ha: μ < 37

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (34 - 37)/(5.5/sqrt(80))
t = -4.879

P-value Approach
P-value = 0
As P-value < 0.05, reject the null hypothesis.

c)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 31
Alternative Hypothesis, Ha: μ ≠ 31

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (34 - 31)/(5.5/sqrt(80))
t = 4.879

P-value Approach
P-value = 0
As P-value < 0.05, reject the null hypothesis.